Bernd Schober: Resolution of Surface Singularities
Nowadays, there are several quite accessible accounts for Hironaka's theorem of resolution of singularities for algebraic varieties over fields of characteristic zero. In contrast to this, there exist only results in small dimensions for the case of positive or mixed characteristic.
In my talk I will briefly recall the statement of embedded resolution of singularities and explain some of the key techniques which lead to a proof in characteristic zero. After this I will point out why these can not be applied in positive characteristic, in general. Based on a result by Hironaka for excellent hypersurfaces of dimension two Cossart, Jannsen and Saito (CJS) gave a proof for resolution of singularities of two dimensional excellent schemes via blowing ups in regular centers. More precisely, they introduced a canonical strategy for the choice of the centers and showed by contradiction that the constructed sequence of blowing ups can not be infinite. After explaining their strategy, I will discuss the case of affine plane curves.
Then, I will outline how to obtain the ingredients for an invariant that captures the strict improvement of the singularity along the CJS process. Therefore, the invariant provides the basis for a direct proof of the result by CJS. The constructions involve Hironaka's characteristic polyhedron which is a certain minimal projection of the Newton polyhedron. Hence, the ideas for the very technical proof of the improvement, can be explained by drawing rather simple pictures.
This is joint work with Vincent Cossart.
